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A324841
Matula-Goebel numbers of fully recursively anti-transitive rooted trees.
3
1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 21, 23, 25, 27, 31, 32, 35, 49, 51, 53, 57, 59, 63, 64, 67, 73, 77, 81, 83, 85, 95, 97, 103, 115, 121, 125, 127, 128, 131, 133, 147, 149, 153, 159, 161, 171, 175, 177, 187, 189, 201, 209, 217, 227, 233, 241, 243, 245
OFFSET
1,2
COMMENTS
An unlabeled rooted tree is fully recursively anti-transitive if no proper terminal subtree of any terminal subtree is a branch of the larger subtree.
EXAMPLE
The sequence of fully recursively anti-transitive rooted trees together with their Matula-Goebel numbers begins:
1: o
2: (o)
3: ((o))
4: (oo)
5: (((o)))
7: ((oo))
8: (ooo)
9: ((o)(o))
11: ((((o))))
16: (oooo)
17: (((oo)))
19: ((ooo))
21: ((o)(oo))
23: (((o)(o)))
25: (((o))((o)))
27: ((o)(o)(o))
31: (((((o)))))
32: (ooooo)
35: (((o))(oo))
49: ((oo)(oo))
51: ((o)((oo)))
53: ((oooo))
57: ((o)(ooo))
59: ((((oo))))
63: ((o)(o)(oo))
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
fratQ[n_]:=And[Intersection[Union@@Rest[FixedPointList[Union@@primeMS/@#&, primeMS[n]]], primeMS[n]]=={}, And@@fratQ/@primeMS[n]];
Select[Range[100], fratQ]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 17 2019
STATUS
approved